Engineering Optimization: Theory and Practice, Fourth Edition

1.5 Classification of Optimization Problems 19

Thus the problem can be stated as an OC problem as

FindX=











x 1

x 2

x 12











whichminimizes

f (X)=

∑^12

i= 1

ti=

∑^12

i= 1















−vi+

√

vi^2 + 2 s

(

xi

mi

−g−

k 1 vi

mi

)

xi

mi

−g−

k 1 vi

mi















subjectto

mi+ 1 =mi−k 2 s, i= 1 , 2 ,... , 12

vi+ 1 =

√

vi^2 + 2 s

(

xi

mi

−g−

k 1 vi

mi

)

, i= 1 , 2 ,... , 12

|xi| ≤Fi, i= 1 , 2 ,... , 12

v 1 =v 13 = 0

1.5.4 Classification Based on the Nature of the Equations Involved

Another important classification of optimization problems is based on the nature of
expressions for the objective function and the constraints. According to this classi-
fication, optimization problems can be classified as linear, nonlinear, geometric, and
quadratic programming problems. This classification is extremely useful from the com-
putational point of view since there are many special methods available for the efficient
solution of a particular class of problems. Thus the first task of a designer would be
to investigate the class of problem encountered. This will, in many cases, dictate the
types of solution procedures to be adopted in solving the problem.

Nonlinear Programming Problem. If any of the functions among the objective and
constraint functions in Eq. (1.1) is nonlinear, the problem is called anonlinear pro-
gramming(NLP)problem. This is the most general programming problem and all other
problems can be considered as special cases of the NLP problem.

Example 1.3 The step-cone pulley shown in Fig. 1.10 is to be designed for trans-
mitting a power of at least 0.75 hp. The speed of the input shaft is 350 rpm and the
output speed requirements are 750, 450, 250, and 150 rpm for a fixed center distance
ofabetween the input and output shafts. The tension on the tight side of the belt is to
be kept more than twice that on the slack side. The thickness of the belt istand the
coefficient of friction between the belt and the pulleys isμ. The stress induced in the
belt due to tension on the tight side iss. Formulate the problem of finding the width
and diameters of the steps for minimum weight.